Theorem imdistan

Description: Distribution of implication with conjunction. (Contributed by NM, 31-May-1999) (Proof shortened by Wolf Lammen, 6-Dec-2012)

	Ref	Expression
Assertion	imdistan	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜒 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pm5.42	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜒 ) ) ) )
2		impexp	⊢ ( ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜒 ) ) ↔ ( 𝜑 → ( 𝜓 → ( 𝜑 ∧ 𝜒 ) ) ) )
3	1 2	bitr4i	⊢ ( ( 𝜑 → ( 𝜓 → 𝜒 ) ) ↔ ( ( 𝜑 ∧ 𝜓 ) → ( 𝜑 ∧ 𝜒 ) ) )